Convexification with bounded gap for randomly projected quadratic optimization

TL;DR

This paper introduces a novel method using random projection to convexify non-convex quadratic optimization problems, enabling approximate global solutions while reducing problem size.

Contribution

It is the first work to leverage random projection for convexification of non-convex optimization problems, providing a new approach to approximate global solutions.

Findings

The convexified problem provides a close approximation to the original non-convex problem.

Random projection reduces problem size while alleviating non-convexity.

The method offers a new way to solve large-scale non-convex quadratic problems.

Abstract

Random projection techniques based on Johnson-Lindenstrauss lemma are used for randomly aggregating the constraints or variables of optimization problems while approximately preserving their optimal values, that leads to smaller-scale optimization problems. D'Ambrosio et al. have applied random projection to a quadratic optimization problem so as to decrease the number of decision variables. Although the problem size becomes smaller, the projected problem will also almost surely be non-convex if the original problem is non-convex, and hence will be hard to solve. In this paper, by focusing on the fact that the level of the non-convexity of a non-convex quadratic optimization problem can be alleviated by random projection, we find an approximate global optimal value of the problem by attributing it to a convex problem with smaller size. To the best of our knowledge, our paper is the first to use random projection for convexification of non-convex optimization problems. We evaluate the approximation error between optimum values of a non-convex optimization problem and its convexified randomly projected problem.